098. Prime Factors and Divisibility Tests

098r. Prime Factors and Divisibility Tests

Learning Intentions

• Solve the prime factor form of a number
• To understand how the lowest common multiple and highest common factor of two numbers can be found Use their prime factor form
• Use the divisibility tests for single digit factors other than 7

Pre-requisite Summary

• Know that a prime number has exactly two factors. See 013. Prime Numbers and Prime Factorisation
• Be able to Identify common prime numbers such as
• Know that a composite number can be broken into prime factors
• Understand that a factor divides a number exactly
• Know the meaning of lowest common multiple and highest common factor. See [[LCM)](HCF|[LCM|011. Highest Common Factor (HCF|[HCF|[HCF|[LCM|[HCF|[HCF|[HCF|[LCM|[LCM|LCM)]]]]]]) and Lowest Common Multiple (LCM)]]%20and%20Lowest%20Common%20Multiple%20(LCM))%20and%20Lowest%20Common%20Multiple%20(LCM)%5C)
• Be able to list simple multiples and factors of whole numbers
• Understand that divisibility tests help decide quickly whether one number divides another exactly
• Recall divisibility tests for small numbers from place value and digit sums

Worked Examples

Worked Example 1

Write each number in prime factor form:

a)

b)

c)

Worked Example 2

Write each number in prime factor form using index notation:

a)

b)

Worked Example 3

Use prime factor form to find the highest common factor of:

a) and

b) and

Worked Example 4

Use prime factor form to find the lowest common multiple of:

a) and

b) and

Worked Example 5

Use divisibility tests to decide whether each number is divisible by or :

a)

b)

c)

Worked Example 6

Use divisibility tests to list all single-digit factors other than that divide each number:

a)

b)

c)

Problems

Problem 1

Write each number in prime factor form:

a)

b)

c)

Problem 2

Write each number in prime factor form using index notation:

a)

b)

Problem 3

Use prime factor form to find the highest common factor of:

a) and

b) and

Problem 4

Use prime factor form to find the lowest common multiple of:

a) and

b) and

Problem 5

Use divisibility tests to decide whether each number is divisible by or :

a)

b)

c)

Problem 6

Use divisibility tests to list all single-digit factors other than that divide each number:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Write each number in prime factor form:

a)

b)

c)

d)

Exercise 2.

Write each number in prime factor form using index notation:

a)

b)

c)

Exercise 3.

Find the highest common factor using prime factor form:

a) and

b) and

c) and

Exercise 4.

Find the lowest common multiple using prime factor form:

a) and

b) and

c) and

Exercise 5.

Use divisibility tests to decide whether each number is divisible by or :

a)

b)

c)

Exercise 6.

List all single-digit factors other than that divide each number exactly:

a)

b)

c)

Exercise 7.

Complete each statement:

a) A number is divisible by if …

b) A number is divisible by if …

c) A number is divisible by if …

d) A number is divisible by if …

Exercise 8.

For each pair, find both the HCF and the LCM using prime factor form:

a) and

b) and

c) and

Reasoning

Exercise 9.

Explain why and represent the same prime factor form.

Exercise 10.

A student says the HCF of two numbers is found by multiplying all the prime factors in both numbers together. Explain the mistake.

Exercise 11.

A student says the LCM of and is because these are common prime factors. Explain why this is incorrect.

Exercise 12.

Without dividing fully, explain why is divisible by and but not by .

Exercise 13.

A student says that if a number is divisible by , then it must also be divisible by . Is the student correct? Explain.

Problem-solving

Exercise 14.

Two bells ring every minutes and every minutes. If they ring together now, after how many minutes will they next ring together?

Exercise 15.

Two teams have and players. What is the greatest number of equal groups that can be made so that each group has the same number of players and no players are left over?

Exercise 16.

A factory packs juice boxes into trays of , cartons of , and crates of . Which of these pack sizes can be used exactly for juice boxes? Use divisibility tests.

Exercise 17.

A school is arranging red counters and blue counters into identical piles with no counters left over. What is the greatest number of counters in each pile if each pile must have the same composition?

Exercise 18.

A number is divisible by and , but not by or . Write two possible three-digit numbers and justify your choices using divisibility tests.

Exercise 19.

Two traffic lights change every seconds and every seconds. After how many seconds will they change together again?

Potential Misunderstandings

• Students may stop factorising before all factors are prime
• Students may include as a prime factor
• Students may write prime factor form in a different order and think it is incorrect
• Students may confuse prime factor form with a list of all factors
• Students may think the HCF uses all prime factors from both numbers instead of only the common prime factors with the lowest powers
• Students may think the LCM uses only common prime factors instead of all prime factors with the highest powers
• Students may mix up when to Choose the lowest power and when to choose the highest power
• Students may think divisibility by only requires divisibility by or , instead of both
• Students may test divisibility by using the whole number instead of the last two digits
• Students may test divisibility by using the whole number instead of the last three digits
• Students may think divisibility by depends on the sum of the digits
• Students may try to Apply a divisibility test for even though it is not part of this lesson

099r. Reviewing Integers and Integer Operations